From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

Owen Barrett (), Frank W. K. Firk (), Steven J. Miller () and Caroline Turnage-Butterbaugh ()
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Owen Barrett: University of Chicago
Frank W. K. Firk: Yale University
Steven J. Miller: Williams College
Caroline Turnage-Butterbaugh: Duke University

A chapter in Open Problems in Mathematics, 2016, pp 123-171 from Springer

Abstract: Abstract The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery’s work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.

Keywords: Compound Nucleus; Critical Line; Riemann Zeta Function; Random Matrix Theory; Riemann Hypothesis (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-32162-2_2

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DOI: 10.1007/978-3-319-32162-2_2

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